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For any integers $x,y,$ we say that $x$ divides $y$ iff there is some integer $z$ such that $y = x\ast z.$
Let $[N, \leq]$ is a partial order relation defined on natural numbers, where “$\leq$” is the “divides” relation defined on $N = { 0,1,2,3,\dots }$ i.e. $a\leq b$ if and only if $a$ divides $b.$ Which of the following statements is/are false ?

  1. $[N, \leq]$ is distributive but not complemented lattice
  2. $[N, \leq]$ is not reflexive.
  3. $[N, \leq]$ is not Boolean lattice
  4. $[N, \leq]$ Element $0$ doesn't have complement
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1 Answer

5 votes
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$[N, \leq]$ is a distributive lattice as $\text{LCM, GCD}$ distribute over each other.

Note that $0$ divides $0$ as $0 = 5*0.$

It is a bounded lattice with $1$ as the least, $0$ as the greatest element, as everyone divides $0,$ and $1$ divides everyone.
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